A Google I/O 2011 talk, "Taking Android to Work", covers this problem, but does not offer a very satisfying solution. See the talk on YouTube (between about 30:00 and 36:00), or the PDF Slides.
In summary, the talk suggests building an "Internal App Directory", which is essentially a private app market for an organisation. The PackageManager API can be used for querying the installed applications, and installing new applications. You'll have to periodically check for updates yourself.
I would think this is a common problem, but unfortunately I haven't found any good existing solutions for this, and it seems like you'll need to implement all of this yourself (both client-side and server-side). You can check out this question, but I didn't find the listed solutions appealing.
If the organisation only uses a single organisation-wide Android application, it is probably simpler to implement all of this in the application itself, instead of in an "Internal App Directory".
Update 2012/03/23:
Since my first answer I checked out some mobile device management (MDM) solutions. Two of them I considered were AirWatch and MaaS360. Both of them have built-in enterprise app stores, which you can use to distribute the apps. The largest issue with them is the pricing - AirWatch is the cheapest decent MDM solution I've found, at $3/user/month.
If the enterprise wants full MDM functionality (many enterprises do), and can afford it, go for an MDM solution with a built-in enterprise app store. If you only want updates for a single application, Pushlink as posted by Victor Hugo seems perfect.
